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Full text of "Treatise On Analysis Vol-Ii"

10   CONVOLUTION OF TWO FUNCTIONS       289

Consequently,

!(/ * 0)00 - (/ * 0)00! ^ NJ0) f I/O""1) - /Oc'5"1)! iv(5)

= NJ0) f (/(a"1) - /(*'*- V1)! dv(s).

The result will therefore be a consequence of the following more general
lemma:

(14.10.6.3)   For p = 1 or p = 2, eiwy n#/tf //oar measure v on G

function h e JS?£(G, v), fAe mapping s\-*§(s)h ofG into JS?£(G, v) w continuous

and satisfies Np(5(j)A) = Np(/z).

The second assertion is an immediate consequence of the right-invariance
of v. To prove the first, suppose first of all that h e 0^C(G) ; then the continuity
of si-*&(s)h follows from (14.1 .5.5). In the general case, if (hn) is a sequence of
functions in tfc(G) which converges to h in ,£?g(G, /?) (13.11.6), the rela-
tion Np(6(^)/z - $(f)hn) = Np(/i - /*„) shows that the sequence of functions
st~+$(s)hn converges uniformly on G to the function st-+&(s)h. Hence the
result (7.2.1).

(1 4.1 0.6.4)   In the same way, one shows that if h e J$?£(G, /?) where;? = 1 or 2,
the mapping s\-+ j(s)h of G into ^(G, /?) is continuous, and that Np(y(/)/z) =

(14.10.7)   Let  fe&l(G,p)   and   let   ge£%(G,p).    Then   the   integral
g(s~~ix)(f(s) d/3(s) is defined for all x e G, and the function f*g belongs to
#c(G) and satisfies the inequality

\

(14.10.7.1)                          \\f*g\\ :

For each x e G, the function s\-+g(s~lx) belongs to JS?c(G, /?), and there-
fore the first assertion follows from (13.11.7). Moreover, again from (13.11.7),

r                        /r      ,      \i/2/r      .   .      \i/2

\\g(s'lx)f(s)\dft(s)^n\f(5)\2t a right Haar measure (14.3.4), and we